MathDB
What quadrilaterals have all these conditions?

Source: Science ON 2021 grade IX/4

March 8, 2021
geometry

Problem Statement

<spanclass=latexbold>(a)</span><span class='latex-bold'>(a)</span> On the sides of triangle ABCABC we consider the points MBCM\in \overline{BC}, NACN\in \overline{AC} and PABP\in \overline{AB} such that the quadrilateral MNAPMNAP with right angles MNA\angle MNA and MPA\angle MPA has an inscribed circle. Prove that MNAPMNAP has to be a kite. <spanclass=latexbold>(b)</span><span class='latex-bold'>(b)</span> Is it possible for an isosceles trapezoid to be orthodiagonal and circumscribed too?
(Călin Udrea)