In figure 3.59, altitudes YZ and XT of \(\triangle WXY\) intersect at P. Prove that,
(1) □WZPT is cyclic. (2) Points X, Z, T, Y are concyclic.
Figure: Z is the foot of the altitude from Y on side WX, and T is the foot of the altitude from X on side WY.
Answer: (1) \(\angle WZP + \angle WTP = 180^\circ\); (2) seg XY subtends equal right angles at Z and T on the same side of it.
(1) YZ ⊥ WX, so \(\angle WZP = 90^\circ\). XT ⊥ WY, so \(\angle WTP = 90^\circ\).
\[\angle WZP + \angle WTP = 180^\circ\]In □WZPT these are opposite angles, so by the converse of the cyclic quadrilateral theorem, □WZPT is cyclic.
(2) \(\angle XZY = 90^\circ\) and \(\angle XTY = 90^\circ\). Points Z and T lie on the same side of line XY, and seg XY subtends equal angles at them.
So by the converse of the theorem of angles inscribed in the same arc, points X, Z, T, Y are concyclic (they lie on the circle with diameter XY).
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