Question

In the figure circles with centres $X$ and $Y$ touch each other at point $Z$. A secant passing through $Z$ intersects the circles at points $A$ and $B$ respectively . Prove that, radius $XA||radiusYB$

Answer 1

Construction: Draw segments $XZ$ and $YZ$.

Proof: By theorem of touching circles, points X, Z, Y are collinear.

$\therefore \angle XZA\cong \angle BZY$ (opposite angles)

Let $\angle XZA=\angle BZY=a.....\left(I\right)$

Now, $segXA\cong segXZ$........(Radii of circle with centre X)

$\therefore \angle XAZ=\angle XZA=a......$(isosceles triangle theorem) (II)

Similarly, $segYB\cong segYZ.......$(Radii of circle with centre Y)

$\therefore \angle BZY=\angle ZBY=a.....$(isosceles triangle theorem) (Ill)

$\therefore from\left(I\right),\left(II\right),\left(Ill\right)$,

$\angle XAZ=\angle ZBY$

$\therefore radiusXAIlradiusYZ......$(Alternate angle test)