Question

Two tangents at A and B cut a third tangent at X and Y. If O is the centre of the circle and angle XOY is equal to 90°, show that the tangents at A and B are parallel.

Answer 1

Answer on Question #46548 – Math - Geometry

Two tangents at A and B cut a third tangent at X and Y. If O is the centre of the circle and angle XOY is equal to $90{}^{\circ}$, show that the tangents at A and B are parallel.

Solution.

$$\Delta AXO = \Delta PXO, \quad \Delta BYO = \Delta PYO$$ $$\begin{array}{l} \text{So, } < AOX > = < POX, < BOY > = < POY, \\ < AOB > = < AOX + < POX + < BOY + < POY > = 2(< XOP + < YOP) = \\ = 2 < XOY = 2 * 90{}^{\circ} = 180{}^{\circ}. \end{array}$$

Thus $< AOB = 180{}^{\circ}$ and the tangents at A and B are parallel.

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