# Triangles and Circles

Theorems on Circles and Triangles including a proof of the Pythagoras Theorem

**Contents**

## Statements Of Some Theorems On The Circle.

**Theorem A**

A straight line drawn from the

**centre**of a

**circle**to bisect a chord which is not a diameter, is at right angles to the

**chord**.

A

**chord**is a**line**segment joining two points on a**circle**.**Theorem B**

There is

**only one circle**which passes through

**three**given

**points**which are not in a straight line.

**Theorem C**

Equal

**chords**of a

**circle**are

**equidistant**from the centre and visa versa.

A

**tangent**is a line intersecting the**circle**at only**one**point.**Theorem D**

The

**tangent**to a

**circle**and the radius through the point of contact are

**perpendicular**to each other.

**Theorem E**

The

**angle**which an

**arc**of a circle subtends at the

**centre**is double that which it subtends at any point on the remaining part of the circumference.

**Theorem F**

**Angles**in the same

**segment**of a circle are

**equal**.

**Theorem G**

The angle in a semicircle is a right angle.

**Theorem H**

The

**opposite**angles of any

**quadrilateral**inscribed in a circle are

**supplementary**.

**Theorem I**

If a

**straight**line

**touches**a circle and from the point of contact a chord is drawn, the angles which this tangent makes with the chord are equal to the angles in the alternate segment.

Example:

##### Example - Supplementary

Problem

To Prove

A line parallel to the base of the triangle cuts , at and respectively. The circle which passes through and touches at meets at . Prove that ,,,, lie on a circle.

A line parallel to the base of the triangle cuts , at and respectively. The circle which passes through and touches at meets at . Prove that ,,,, lie on a circle.

Workings

Construction

Draw the line

Draw the line

**Proof**Since is parallel to the angles and are**supplementary**Hence the angles and are supplementarySolution

Conclusion

from

from

**Theorem H**above the points ,,,, lie on a circle.## Statements Of Some Theorems On Proportions And Similar Triangles.

**Theorem J**

If a straight line is drawn parallel to one side of a triangle, the other two sides are divided proportionally.

**Theorem K**

If two triangles are equiangular their corresponding sides are proportional.

**Theorem L**

If two triangles have one equal angle and the sides about these equal angles are proportional, then the triangles are similar.

**Theorem M**

If a triangle is drawn from the right angle of a right angled triangle to the hypotenuse, then the triangles on each side of of the perpendicular are similar to the whole triangle and to one another.

**Theorem N**

The internal bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the angle.

Example:

##### Example - Example 1

Problem

Any line parallel to the base of a triangle cuts and in and respectively. is any point on on a line through parallel to . If and produced cut at and respectively, Prove that = .

Workings

**Proof**Since AP is parallel to BC Hence triangle BHQ is equiangular with triangle APH Likewise the triangles APK and KCR are equiangular and hence:- Since HK is parallel to BC, theorem (K) applies and we can write:- Combining the three above equations:-

Solution

Conclusion

It can be seen that =

It can be seen that =

## Pythagoras Theorem

Theorem m provides a convenient method of proving Pythagoras theorem.
Consider the right angled triangle .

**Pytagoras theorem**

Given the right triangle prove that .

**Construction**.

Draw such that the angle is a right angle.

**Proof**

The triangles : and are equiangular and similar. From therefore From therefore Add equations (22) and (24) therefore

**Note**

means that the two triangles are similar .

## Two Theorems On Similar Rectilinear Figures.

Polygons which are equiangular and have their corresponding sides proportional are said to be similar. If also their corresponding sides are parallel, they are said to be

**similarly situated (or homothetic)****Theorem**

**1**The ratio of the areas of similar triangles (or polygons) is equal to the ratio of the squares on corresponding sides.

**Theorem**

**2**If O is any fixed point and ABCD.....P is any polygon and if points A'B'C'.....P' are taken on OA,OB,OC,....OP ( or these lines produced in either direction) such that :

**centre of similitude**of the two polygons. If the corresponding points of the two polygons lie on the same side of O the Polygons are said to be {directly homothetic} with respect to O and O is said to be the external centre of similitude. If the corresponding points lie on opposite sides of O then the Polygons are said to be

**inversely homothetic**with respect to O and O is called the internal centre of similitude.

Example:

##### Example - Similarities

Problem

is an acute angle triangle. Show how to construct a square with two vertices on and one vertex on and one on .

Workings

Draw a square on the other side of to the triangle. Join and and let these lines meet at the points and respectively. Draw and perpendicular to to meet and at the points and respectively.

Solution

Then is the required square for regarding as the centre of Similitude is similar to and is therefore a square.