91Ó°ÊÓ

If quadrilateral PQRS is inscribed in \(\mathrm{ON}\) and \(\mathrm{m} \angle \mathrm{Q}=80^{\circ}\), find \(\mathrm{m} \angle \mathrm{S}\).

Prove that the sum of the alternate angles of any hexagon inscribed in a circle is equal to four right angles.

Given: Regular hexagon ABCDEF. Prove: Quadrilateral \(\mathrm{ABDE}\) is a rectangle.

Find the length of an arc intercepted by a side of a regular hexagon inscribed in a circle whose radius is 18 in.

Prove that if two diagonals of a regular pentagon intersect each other, the longer segments will be equal in length to the sides of the pentagon.

An equilateral triangle and a regular hexagon are inscribed in the same circle. Prove that the length of an apothem of the hexagon is greater than the length of an apothem of the equilateral triangle.

Prove: (a) If a triangle has two unequal angles, the smaller angle has the longer angle bisector (measured from the vertex to the opposite side). (b) Any triangle that has two equal angle bisectors Is Isosceles. (The Steiner - Lehmus Theorem).

Prove that if a quadrilateral is inscribed in a circle, the product of the diagonals is equal to the sum of the products of both pairs of opposite sides. (Hint: Consider \(\underline{A E}\) such that \(\angle \mathrm{DAE} \cong \angle \mathrm{CAB}\).)

The product of two sides of a triangle is equal to the product of the altitude to the third side and the diameter of the circumscribed circle. Prove this. (Hint: consider the diameter that passes through the included vertex of the first two sides.)

Using Ptolemy's Theorem show that if \(\mathrm{a}\) and \(\mathrm{b}\), with \(\mathrm{a} \geq \mathrm{b}\), are chords of two arcs of a circle of unit radius, then \(\mathrm{d}=(\mathrm{a} / 2)\left(4-\mathrm{b}^{2}\right)^{1 / 2}-(\mathrm{b} / 2)\left(4-\mathrm{a}^{2}\right)^{1 / 2}\) is the chord of the difference of the two arcs.