Theorems in geometry

Theorem :- Basic Proportionality Theorem (BPT) 
If a line parallel to one side of a triangle interests the other two sides in two distinct points,  then the line divides the other two sides in proportion.

Theorem :- Converse of Basic Proportionality Theorem
If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

Theorem  of Pythagoras :-
In a right angled triangle ,  the  square  of the hypotenuse is equal to the sum of the squares of the other two sides.

Converse of Pythagoras Theorem :-
In a triangle ,  if the square of one side  is equal to the sum of the squares of the remaining two sides , then the angle opposite to the first side is a right angle and the triangle is a right angled triangle.

Theorem of 30° 60° 90°   Triangle
  If the angles of a triangle are 30°, 60° and 90°, then the side opposite to 30° is half of the hypotenuse and the side opposite to 60° is
√3/2 times the hypotenuse

Converse of 30° 60° 90° triangle
In a right angled triangle,  if the length of one side is √3/2 times the hypotenuse, then the measure of an Angle opposite to that side is 60°.

Theorem of 45° -  45° -  90° triangle
If the angles of a triangle are 45°, 45° and 90°, then each of the perpendicular sides is 1/√2 times  the  hypotenuse .

              Theorem used in Circle

Theorem : A tangent at any point of a circle is perpendicular to the radius through the point of contact.
Converse :  The line perpendicular to a radius at its outer end is a tangent to the circle.

Theorem : The lengths of the two tangent - segments from an external point to a circle are equal.

Theorem : If two circles are touching  circles, then the common point lies on the line joining their centres.

Inspired Angle Theorem :
Theorem : The measure of an angle subtended by an arc at a point on a circle is half of the measure of the angle subtended by the  arc at the centre.

Theorem : The  opposite  angles of a cyclic  quadrilateral are supplementary.

Converse : (test for a quadrilateral to be cyclic )
    If a pair of opposite  angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.

Theorem : If an Angle with its vertex on the circle whose one side touches the circle and the other interests the circle in two points, then the measure of the angle is half the measure of the interested arc.

Theorem : If a secant is drawn through the point of contact of a tangent to a circle, then the angles which tangent make with the chord contained in the secant are equal respectively to the angles subtended by the  chord in the corresponding alternate segments.

Converse : If a line is drawn interesting  a secant of a circle at a common point of the secant and the circle so that the angle formed by it with the chord contained in the  secant is equal to the angle subtended by the chord in the alternate segment, then the line is a tangent to the circle.