# Triangles and Circles

**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**.

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**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.

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**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.

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**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.

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**Theorem F**

**Angles**in the same

**segment**of a circle are

**equal**.

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**Theorem G**

The angle in a semicircle is a right angle.

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**Theorem H**

The

**opposite**angles of any

**quadrilateral**inscribed in a circle are

**supplementary**.

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**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.

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##### Example - Supplementary

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.

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Draw the line

**Proof**Since is parallel to the angles and are

**supplementary**Hence the angles and are supplementary

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.

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**Theorem K**

If two triangles are equiangular their corresponding sides are proportional.

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**Theorem L**

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

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**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.

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**Theorem N**

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

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##### Example - Example 1

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**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:-

It can be seen that =

## Pythagoras Theorem

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**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.

**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.

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**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 :

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**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 - Similarities

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