Consider a point P on the Euler circle c(E, ED) of triangle A1A2A3. Define the symmetric I of A1 w.r. to P and show that angle(CA3D) = (1/4)angle(PED), D being the foot of the altitude and A3C being the bisector of angle A1A3I.
(1) Extend A3I and locate its intersection point B with the circumcircle. Show first that PD is parallel to A1B. (2) Use the homothety w.r. to the orthocenter H, mapping the Euler circle to the circumcircle and define the angle(MON) homothetic to anlge(PED). Compare angle(MON) and angle(CA3N).
This property is used in the discussion (and animation) of all rectangular hyperbolas circumscribed on the triangle A1A2A3. Such a hyperbola is uniquely determined by its center which is a point (P) of the Euler circle. See the file RectHypeCircumscribed.html for the details.
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A property of the Euler circle ![[0_0]](EulerCircleProperty_files/EulerCircleProperty_0_0_0.png)
![[0_1]](EulerCircleProperty_files/EulerCircleProperty_0_0_1.png)