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For $i=1,2$, let:

$\Gamma_i$ two circles intersecting each other at $A,B$.

$r$ a line containing $A$ intersecting $\Gamma_i$ at $T_i\neq A$.

$t_i$ tangent line to $\Gamma_i$ at $T_i$.

$P=t_1\cap t_2$.

Prove that the quadrilateral $PT_1BT_2$ is inscribed in some circle.

I'm trying to prove that two adjacent internal angles determined by sides and diagonals are congruent. But no success yet.

Any hint?

Sigur
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1 Answers1

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Extend $\overline{PT_i}$ to a point $Q_i$. Then $\angle BT_iQ_i$ subtends the same chord ($\overline{BT_i}$) as $\angle BAT_i$; these angles must be congruent (by the tangent-secant variant of the Inscribed Angle Theorem). Since $\angle BAT_1$ and $\angle BAT_2$ are supplementary, we have $\angle BT_1P \cong BAT_2$ and $\angle BT_2P \cong \angle BAT_1$. Therefore, $\square PT_1BT_2$, with supplementary angles at $T_1$ and $T_2$, must be cyclic. $\square$

It's worth noting that the angles at $P$ and $B$ are right angles.

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