∠QAB = ∠RBA ; ∠QAB = ∠ACB
∠RBA =∠ACB ; ∠SBC = ∠UCB
∠SBC = ∠BAC ; ∠UCB = ∠BAC
∠ACT = ∠PAC ; ∠ACT = ∠ABC
∠PAC = ∠ABC
Tangents Memory Map
Tangent to the circle is perpendicular to the radius through the point of tangency.
The tangents from an exterior point to a circle and radii to the points of tangency form a cyclic quadrilateral. In figure PAOB is a cyclic quadrilateral.
Tangents from an exterior point to a circle are equal. If PA, PB are the tangents then PA = PB.
The angle between a chord of a circle and the tangent at one end of the chord is equal to angle formed by the chord in the other side of the circle.
If a circle touches the sides of a quadrilateral, that circle will be the incircle of that quadrilateral. Sum of the opposite sides of such quadrilateral are equal. In the figure, ABCD is a quadrilateral having incircle AB + CD = AD + BD.
If P is an exterior point to a circle, a line from P touches the circle at T on the circle and a line intersects the circle at A and B then PA × PB = PT2.
The center of the circle which touches two lines will be a point on the bisector of the angle between the lines. The bisectors of the angles of a triangle passes through a point. That point will be the incenter of the triangle.
The circle drawn inside a triangle which touches the sides of the triangle is called incircle. The circle drawn outside the triangle which touches the sides of the triangle are excircles.
The radius of the incircle of a triangle is obtained by dividing area of the circle by its semi perimeter.
If a, b, c are the sides of a triangle then the area of the triangle A = \(\sqrt{S(S-a)(S-b)(S-c)},\)
\(s=\frac{a+b+c}{2}\)
This is popularly known as Hero’s formula.
