Question 2
A fixed circle has center $O$ and radius $r$. A variable triangle $ABC$ always circumscribes this circle, with vertex $A$ on a fixed line $x$ and the other two vertices on a fixed line $y$ parallel to $x$.
(a) Given the circle $O$, the lines $x,y$, and the angle $A$, construct triangle $ABC$. Discuss.
(b) Calculate the angles of triangle $ABC$ in terms of angle $A$, radius $r$, and the distance $h$ between $x,y$. Discuss.
(c) What relationship do you notice between $IB$ and $IC$ ($I$ is the tangency point of circle $O$ with line $y$)? Is there a way to guess this relationship without proving anything yet? Prove the conjecture.
From the relationship between $IB$ and $IC$, try to discover other noteworthy relationships (the more you discover, the better).
(a) Given the circle $O$, the lines $x,y$, and the angle $A$, construct triangle $ABC$. Discuss.
(b) Calculate the angles of triangle $ABC$ in terms of angle $A$, radius $r$, and the distance $h$ between $x,y$. Discuss.
(c) What relationship do you notice between $IB$ and $IC$ ($I$ is the tangency point of circle $O$ with line $y$)? Is there a way to guess this relationship without proving anything yet? Prove the conjecture.
From the relationship between $IB$ and $IC$, try to discover other noteworthy relationships (the more you discover, the better).