Problem: a, b, c are lengths of side of a triangle

Let a, b, c are lengths of three sides of a triangle, and:

Show that: P < 0

Solution:

We can assume that c this the longest side of the triangle ABC (it does not make different if the longest side is a or b), draw the height CH with H is on AB => H is lying between points A and B (see the pic below)

then: 

let change the expression P as following:

Geometry problem (2)

Given triangle ABC, angle BAC < 90 degrees. Two squares with side AB, AC are drawn outside of the triangle. M is a midpoint of BC. Show that DM ⊥ ME and DM = ME

Geometry Problems (1): Prove that points are laying on one circle!

Given arbitrary five-wings star, the circumscribed circles of its wings are drawn. Show that intersections points A, B, C, D, E are laying on one circle!

Geometry Pad-2D: Select an object to work with

In many cases you need to select an object (point, line segment, circle, triangle, polygon, angle) to work with. For example, you want to draw a circumscribed circle of triangle ABC, in this case you need to select triangle, and use triangle tools to draw its circumscribed circle.

There are some ways to select an object in the construction showing in screen:

Select by tapping on screen

• To select a point, you can simply tap nearly it on screen
• To select a line segment: tap nearly its center point
• To select a circle: tap on it
• To select a triangle, polygon: tap somewhere in the "centre" of a shape
• To select an angle: select its vertex first, then tap somewhere inside its, nearly but not too close to the vertex     

select an object by tapping on screen

Limitation: when your construction is complex enough, this may make some confuse when you tap on the screen. For example, when two or more line segments have the same center point or there are some angles with the same vertex,...

Note: the selected object by default will be highlighted light-green filling

Use "bottom" tools 

This tools will appeared when you select some point. It is to help you select objects related to the selected point. For example, when you select point "B", the "line segment" selection tool will show you all line segments comes from/to point "B".

bottom selection tools

Limitation: this method shows you all possible variations for choosing an objects, sometime there are too many variations and it makes you difficult to detect which one is your object!

Use main selection tools

This is the most careful method to select an object, it was designed for line segments, triangles, angles and polygons. Steps to choose an object are:

1. Choose selection tool
2. Choose type of object
3. Tap on all points (vertices) of an object which you want to select

Note: You should tap on the points in order you needed. For example, when you tap points: "A", "B", "C", angle ABC will be selected; but tapping on "B", "A", "C" will select angle "BAC". In the case of polygon: quadrilaterals ABCD and ACBD are different each of other!

Free download

2014.11.13
Geometry 2D-Pad on Facebook

Geometry Pad-2D: Working with triangle [1]

Suppose that we have some problem like: "Given triangle ABC with the height AH, draw line HD // AC, D is on side AC...."

pic 1

With the "Geometry Pad-2D" app, you can draw the picture easily by following steps:

Step 1: Draw triangle ABC

Step 1: Draw triangle ABC

a) Tap on the "+" tool of the TOP toolsbox
b) Tap on the "triangle" tool of the RIGHT toolsbox
c) Select type of triangle
d) Move triangle into right position 

Note: You also can change the drawn triangle by tap on any vertex or side then move it, change it using points or line segments tools

Step 2: Draw height AH

There are different ways to draw the heights of triangle: use points tools, use lines tools or use triangle tools. The following shows how to use points tools to draw height AH.

a) Select vertex A
b) Select "draw perpendicular segment" tool
c) Select showed point to draw line to...

Note: new the point on BC is automatically named as "D", we will change name of the point later.

Step 2: Draw height from A

Step 3: Draw line segment parallel to  AB

a) Select line AB, then select point D
b) Select "draw perpendicular..." of the Right toolsbox
c) Tap on small brown circle appeared on AC

Step 3: Draw DE // AB

Note: by default, new added line segments have "dotted" style









Step 4: Naming, styles and colours

a) Now you can use points tools (select any point to show point tools on the right side of screen) to change name(s) of points. In our case change "D" -> "H" and "E" -> "D"

(The naming tool is the first one of the points toolsbox)

b) Select line segment to show lines toolsbox on the right side of screen. Use change line-style, change line-colour tools to make a picture look as you want

line segments toolsbox

c) Select angle AHC and mark it with, the right angle mark will be drawn

mark angle

Free download

2014.11.10
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Geometry 2D-Pad: How to add new object?

Free download

This is to describe the first way to add new object to a geometry construction using Geometry 2D-Pad (v2.0) on your iPad (with iOS 7++). In our example, the new object is a triangle.

1) Tap the "add" tool on the top screen menu (+)
2) Tap the triangle tool on the right toolset (it appeared after 1st action)
3) The new triangle with the "moving circle" (pink) will appeared in drawing screen
4) Drag and drop the new shape with the pink circle at the right position

See the picture:

Step to add new triangle

The triangle ABC has been added. Now you can choose it and use "working with triangle toolset" to add the altitude AH as bellow:

Choose triangle ABC and draw altitude AH

Notes: before moving the new triangle, you could see some interesting pictures on the bottom side of drawing area, what is it? This is a new interesting feature, which allows you to choose a type of new triangle: equilateral, right or isosceles. Let try it and see the results!

It is so easy, isn't it?

Free download

2014.10.03
Geometry 2D-Pad on Facebook

How to draw an inscribed and a circumscribed circles of a triangle?

(O1) is an inscribed, (O2) is a circumscribed circle of triangle ABC

This post is to show how to use Geometry Drawer tools to draw an inscribed and a circumscribed circle of a given triangle. On a paper, this task is a combination of several activities to detect a centre points of required circles. With Geometry Drawer Tools it made easy:

1) Select a triangle ABC with selection tools or by selecting point A, then long press on on side BC
2) When the triangle is selected, the circles tools will be appeared.
3) Select required tool and drag it into drawing area, then drop it

That is all! See the picture bellow:

Tools for drawing an inscribed and a circumscribed circles

Notes:

1) The picture is taken by Geometry Drawer for iPhone. It is the same way for drawing by Geometry Drawer for iPad.

2) After the above task, you can use circles tools to change style and colour of a drawn circles

11.6.2014

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Plane Geometry Drawer for iPhone

How to draw perpendicular and parallel lines with points toolset

picture 1

Assume that we have a triangle ABC and we want to draw the height AH and parallelogram ABCD (draw the red lines on the picture 1).

The following steps show how to do that.

Step 1: Draw line AH ⊥ BC

Select point A. Drag the "add line" tool of the point toolset and drag it to line BC

The small green circles appeared to highlight "meaning points", one of them is a projected point of point A on BC (call it H).

Move the red pointer to H and drop. Point A will be connected with H. (Select AH and use color tool to change color of AH to red)

illustration of step 1

Step 2: Draw AD // BC

Select point A, choose "parallel line" tool of the point toolset (//) and drag it into the area above of BC. Some points will be highlighted as shown in picture 3. Move the pointer to the point 2 and drop it there (see picture 3). Name the new point as D.

picture 3

Connect point C with point D and colour lines AD, CD by red. In the result we have a drawing which is shown in picture 1.

Note: There a variety ways to draw the construction. The above is method by using points toolset, we can also use line toolset, circles toolset or combinations of them.

16.3.2014

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Plane Geometry Drawer for iPhone

 

How to start drawing a geometry construction

Imagine your construction

Before starting drawing, it is important to think about the construction: which is the main object? (a triangle? a square? a trapezoid? or a circle?...) and where to start?... Try to imagine your construction and choose the object to start.

Add the first object(s)

P1: Basic objects tools

The drawing screen starts with the basic objects tools. Choose your object, drag into the drawing area and drop it! The most common object to start are triangle, line and circle.

After adding the first object, you can add the second,...

Note: basic objects tools add an object independently of other objects on your constructions. If you want to add an object with some relation with existing object (for example: draw line from a selected point to other point), choose points tools, line tools,... in accordance with your selected object.

P2: connecting points

Draw line connecting a selected point with other point

The picture P2 shows a simple construction with line segment BC and point A. Touch on point A to select it. Now you can use line drawing tool (the first one on the left of toolset) of points toolset to draw line from point A to any other point.

Touch and hold the line tool, drag it into drawing area (do not drop it) and you can see 5 green circles appeared with the pink one connecting to A by a dotted red line (P3).

1) Green circles are points A, B, C, projected point of A on BC and the midpoint of BC.

2) The pink circle is pointer of current position, drag and drop it on the point you want to connect with point A.

P3

For example, we do the following:

- connect A with B (draw line AB)
- connect A with C (draw line AC)
- connect A with midpoint of BC (draw the median AM)
- name the midpoint of BC as M

Finally we have a triangle ABC with the median AM.

Note: when you drag nearly to the highlighted points, the app with automatically adjust the pointer into it. This helps you draw easier and more exactly. 

11.3.2014

Geometry Drawer on Facebook

Plane Geometry Drawer for iPhone

Tools News: Plane Geometry Drawer v1.3 is available on the AppStrore

The new version (1.3) of "Plane Geometry Drawer" has the following new features/improvements:


1) Tools for set colour for lines and circles

When any line/circle is selected, the small circle will appear at the top of the drawing area (on the left). Touch on it to select colour for the selected line/circle.


2) Improvement of the objects selection tools

Now if you touch on the "pink" circle, all options are listed and you could select the objects type by one touch. One more change: for selecting a line, choose the line come from/to a selected point.


FREE Download!

3) New keyboard with more geometry symbols

The new geometry keyboard add more geometry and math symbols and some of greek characters.

4) Naming points

Automatic increase the next points name for complex names like A', B', (O), (I),... For example if you named a point as A', you can name the next point(s) as B', C' by dragging the naming tool for the point(s).




FREE Download!

09.3.2014

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Plane Geometry Drawer for iPhone

Practice: how to draw rectangle and angle

This article shows how to draw the rectangle ABCD, AD = 2AB and point E is on side BC and angle BAE = 15°

picture 1

picture 2

There are different ways to draw a geometry constuction shown in picture 1. The following steps are one of constructing methods:


1) From the basic tools select the "add line" tool and add a line, let name it AD


2) Select point A by touching on it, the points tools will be showed

picture 3

3) Use draw perpendicular line from A, draw AB ⊥ AD and AB = 1/2 AD (picture 2)


4) Select point B, and draw BC // AD, BC = AD, draw line from C to D. Now we have a rectangle ABCD with AD = 2AB.

5) Select point A and from A draw arbitrary line AE, and drag the protractor to point A (picture 3), use protractor to rotate AE such as angle EAD = 75°.

6) Use change length tool to move point E to line BC. In the end we have the construction as required.

7.3.2014

Geometry Drawer on Facebook

Plane Geometry Drawer for iPhone

Drawing with selected line

With the selected (straight) line segment (or "line" in short) the "Plane Geometry Drawer" app supports variety set of users actions:

1) Add a new point on selected line AB (highlighted)
2) Change line style solid <-> dotted
3) Draw a new line perpendicular to the selected line
4) Draw a new line parallel to the selected line
5) Change the length of the line
6) Draw extensions of the line
7) Draw a circle tangent to the line
8) Draw a cicle with selected line as diameter
9) Draw a triangle with selected line as one side and with given angle


With the PRO version you can also mark the selected line with different mark signs (see the picture).

New!!! With the new, coming soon version 1.3, the tools will allow users to choose color for lines & circles. 

06.3.2014

Geometry Drawer on Facebook

Plane Geometry Drawer for iPhone

Drawing with selected point objects

Draw line from A perpendicular to CD

When you tap on any point of construction, the point will be highlighted by the small green circle around it and point tools will be appeared on the bottom of screen.

There many useful tools for drawing with selected points (A). To select a point touch on it or use selection tools (point option):

1) Draw line form point A to other points or new point
2) Draw straight line from point A perpendicular to arbitrary line. A picture shows how to do that.
3) Draw straight line from point A parallel to arbitrary line
4) Draw an arc of center A, with required length on any direction
5) Draw a circle with a diameter from A to any point
6) Draw a circle passing through A and tangent line
7) Draw a circle of center A and passing through any other point
8) Draw a circle passing through A and two other points


Simply drag the tool and drop it where you needed!

06.3.2014

Geometry Drawer on Facebook

Plane Geometry Drawer for iPhone

Users Guides: How to draw basic geometry objects

Drawing basic objects (point, line, circle, angle, triangle, square, ...) is easy task: tap on the respective tool icon, drag it to the drawing area and drop it where you want. But before that you need to show basic tools by tapping on any point on the drawing area (not on existing objects).

Usually, use the basic tools to add an "basic object" or an "main object" or the first object of your construction. 

The tools adds objects independently, the new object doesn't have any relations with previous objects.

To draw objects with relations with existing objects, we use other toolsets: points tools, line tools,....

06.3.2014

Geometry Drawer on Facebook

Plane Geometry Drawer for iPhone

Plane Geometry Drawer v1.2 is now available on the App Store

Tools News: FREE Download

The new version of the drawing tools is now available on the Apple Appstore is running on devices with iOS 7.0 or later. The version provides new useful features:

New tools to select drawing objects on a iPhone/iPod Touch screens: Tools for selecting an object to work with. Now you can choose the needed objects which are closed to others. Usually this is an difficult action:

It is not easy to select objects like: angle AHP or segment BH

In the following screen you can see a small pink circle on the top-left of the drawing area. It will appear when you touch on any point of the drawing picture. There are selection modes for different types of objects: angles, triangles, circles, lines and points. The initial mode is for selecting a line, touch on the pink circle to change the working mode.

Plane Geometry Drawer for iPhone v1.2: Object selection tools (pink circle)

To use selection tools, please follow the steps below:

1) Touch on random point of the drawing construction, pink circle will appeared
2) Pan on the drawing area from any point to move the circle to the required object:
    - In point mode: move close to the point to be selected
    - In line mode: move close to the midpoint of the line
    - In angle mode: move to the midpoint of the line formed by two points on two rays of the angle
    - In triangle mode: Selected one vertex of triangle, then move the pink circle to the midpoint of the opposite side of the needed triangle
    - In circle mode: Move the pink circle to any point near the circle

Plane Geometry Drawer for iPhone v1.2: Selected a line segment OH

Toolset for marking right angles or equal line segments/angles: This feature is available for PRO version only. It helps users to mark equal/not equal angles or line segments. The toolset will appear on the right side of drawing area when any angle or line is selected! Touch on the button to mark the selected objects.

Other improvements: some geometric symbols are added to the geometry text editor keyboard like: ° (degree) or π (Pi) or some tools are added to existing toolsets.

Plane Geometry Drawer for iPhone v1.2: New additional symbols (geometry text editor tools)

FREE Download!

20.2.2014

Geometry Drawer on Facebook

Plane Geometry Drawer for iPhone

Requirements of geometric drawing: The science

Science in the drawing geometry? This topic is somewhat a bit confusing, so to make things simpler we will consider a example.

Example: Given ΔABC, the height AH, median AM (H and M on BC).

Very simple, isn’t it? Then how will we draw such a figure? We can draw as below:

Figure 1: Triangle ABC, the height AH, the median AM

It is absolutely accurate according to the original data. The problem here is that points M and H are too close together, and that will lead to difficulty to distinguish between them and between two lines AH and AM, making simple drawing become unrecognizable.

What if after that, we need to identify the circumcenter and center of gravity of ΔABC? In this case, we can say Figure 1 is "not scientific". In general, it would be a lot better if drawing like Figure 2 below:

Figure 2: Triangle ABC, the height AH, the median AM

Similarly, it can be noticed that Figure 3 or Figure 4 below are not scientific (except for cases that are required to draw as such):

Figure 3: Triangle ABC, the height AH, the median AM

Figure 4: Triangle ABC, the height AH, the median AM

So to draw scientifically, what do we have to note? The following suggestions may be helpful and should be applied when you start drawing (if not bound by other facts):

1) When drawing angles or triangles, try not to draw the vertices with special values, such as 30°, 60°, 45°, 90°, 120°, 180°, ...

2) The edges of the triangles, parallelograms or rectangle should not be drawn approximately equal. For example, if you draw a rectangle ABCD, the lengths of AB and BC should not be close in value and also one should not double the other!

3) If you draw a trapezoid or a triangle, you should not draw an isosceles one. Normally, the two trapezoidal sides should be approximately 1/3 different in lengths. For example, the trapezoid ABCD, AB // CD, you should draw the side BC = about 2/3 or 4/3 of the DA.

4) Use the solid or interrupted lines logically. For example, if two sides of a triangle are in solid lines, the 3rd edge should be also be solid. Or when you have multiple solid lines converge at one point, then an additional line should be an interrupted line...

There may be many other points to note, but the above points have helped a great deal. Most importantly, the figure should be easy to read, easy to understand and highlight the given facts as well as the relationships between them.

Related articles:

Geometric drawing requirements: accuracy

Requirements for geometry drawing: overview

9.2.2014

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Plane Geometry Drawer for iPhone

Geometric drawing requirements: accuracy

Related article(s): Plane geometry: The requirements for the drawing - an overview

In the previous article about an overview of the requirements for the plane geometry drawing mentioned the requirements for accuracy, scientific and aesthetics. This article will further analyse the first requirement: accuracy! Accuracy in drawing is especially important and reflected in the following key points:

1) Angles with a given value as of 90 °, 60 °, 45 °, 30 °, ... need to be accurately constructed. Or an angle bisector must divide that angle into two congruent parts,... Let’s look at the diagram Pic 1: 

Suppose ΔABC = ΔJIK and 2 straight lines AD and IL are the bisector of the angle BAC and JIK respectively. Did you think that both figures were correct?

Pic 1: AD is the bisector of angle BAC and IL is the angle bisector of JIK

Next, draw two circles C1, C2 of center D and L that are tangent to the rays of ∠BAC and ∠JIK respectively! Pic 2 shows what would happen:

Pic 2: Draw a circle of center D, tangent to AB and a circle of center L, tangent to IJ

C1 touches both rays AB and AC while C2 only touches IJ but not IK! The problem is that bisector IL is drawn incorrectly (specifically in this case ∠JIL < ∠LIK). Clearly, although looking at pic 1, both figures seem to be good, the 2nd picture in fact is inaccurate and will not allow us to continue to solve the problem as figure Pic 2.

2) Accuracy is also reflected in the correlation in length of the line segments. For example, given the triangle ΔABC with AC > AB, in the drawing, the length of AC should be greater than AB. If given two straight lines of equal length, then their lengths must be equal in the drawing. Or if you draw two perpendicular lines, the angle created by two lines in the figure must be exactly 90°. For example, take a look at the figure below:

Pic 3: Given ΔABC, angle A = 90°

Which triangle is correctly drawn? Are both or either correct? Given angle A is a right angle, hence if we draw a circle of diameter BC, then A will be one of the points on such circle. We have:

Pic 4: for ΔABC, angle A = 90°, draw a circle of diameter BC

Figure Pic 4 showed that apparently the below triangle in Figure Pic 3 is not correct. And if we continue to draw, the figure would become unreasonable. Those above are just two of many examples showing that if we did not draw accurately, we could not continue to study or solve geometric problems. The accuracy in drawing is absolutely important and is the first requirement!

6.2.2014

Geometry Drawer on Facebook

Plane Geometry Drawer for iPhone