244. Some properties of cyclic orthodiagonal quadrilaterals.

by Virgil Nicula, Mar 5, 2011, 12:16 PM

Remarks. IF $ ABCD$ is an orthodiagonal convex quadrilateral inscribed in the circle $ w = C(O,r)$ , THEN :

$ 0.\ \Longrightarrow$ If denote the midpoint $ M$ of $ [AB]$ and the projection $ X$ of $ P$ to $ BC$ , then $ P\in MX$ a.s.o. analogously.

$ 1.\ \Longrightarrow$ The midpoints of the its sides and the projections of $ P\in AC\cap BD$ to the its sides belong to the circle

with the center in the midpoint of $ OP$ and with the length of radius $ \boxed {\ R = \sqrt {2r^2 - OP^2}\ }$ (the circle of eight points).

$ 2.\ \Longrightarrow\ \sum PA^2 = \boxed {\ AB^2 + CD^2 = AD^2 + BC^2 = 4r^2\ }$ .

$ 3.\ \Longrightarrow\ \boxed {\ AC^2 + BD^2 = 4R^2\ } = 4\cdot \left(2r^2 - OP^2\right)$ .

$ 4.\ \Longrightarrow\ \sum PA = \frac {(AB + CD)(AD + BC)}{2r}$ . Denote $ \delta_d(X)$ - distance of $ X$ to the line $ d$ .

$ 5.\ \Longrightarrow\ \boxed {\ \delta_{AB}(O) + \delta_{BC}(O) + \delta_{CD}(O) + \delta_{DA}(O) = s\ }$ (semiperimeter).
This post has been edited 2 times. Last edited by Virgil Nicula, Nov 22, 2015, 12:24 PM

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