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 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.
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.
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.
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.
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.
Angles in the same segment of a circle are equal.

Theorem G
The angle in a semicircle is a right angle.
The angle in a semicircle is a right angle.

Theorem H
The opposite angles of any quadrilateral inscribed in a circle are supplementary.
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.
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
Workings

Draw the line
Proof
Solution
Conclusion
from Theorem H above the points
,
,
,
, lie on a circle.
from Theorem H above the points
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.
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.
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.
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.
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.
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

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
.
Given the right triangle
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 .
Draw
Proof
The triangles
Note
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 :

Then the polygons ABCD.....P, A'B'C'D'.....P' are similar and similarly situated.

Example:
Example - Similarities
Problem
Workings

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