<s>[INST] Geometry Prove - two perpendicular lines in a circle In a circle of radius r, two lines (AB and CD) are perpendicular to each other and meet at X. 
Show that:


 [/INST] a^2 + y^2 = r^2\\
B: b^2 + y^2 = r^2 \\
C: x^2 + c^2 = r^2 \\
D: x^2 + d^2 = r^2
$$
From $A$ and $B$ one can find that $a = -b$, and analogously $c = -d$. Then
$XA^2 + XB^2 + XC^2 + XD^2 = (a-x)^2 + (b-x)^2 + (c-y)^2 + (d-y)^2 = \\
a^2+b^2+c^2+d^2-2x(a+b)-2y(c+d)+2x^2+2y^2 = a^2+b^2+c^2+d^2+2x^2+2y^2$
which is sum of all LHS for the equations $A-D$ above and consequently $4r^2$
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