Reflections of a Point on the Circumcircle
What Is This About?
A Mathematical Droodle

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Reflections of a Point on the Circumcircle

The applet suggests the following theorem [Coxeter & Greitzer, p. 45, Honsberger, pp. 43-44]:

The reflections of a point on the circumcircle of a triangle in the side lines of the latter are collinear with the orthocenter of the triangle.

Let P be a point on the circumcenter of ΔABC. T_a (T_b, T_c) is the foot of the perpendicular from P to BC and P_a (P_b, P_c) is the reflection of P in BC (AC, AB). We wish to prove that the four points P_a, P_b, P_c, and H are collinear.

Note that the points T_a, T_b, T_c lie on the simson of P with respect to ΔABC. Also, the points P_a, P_b, P_c are the images of points T_a, T_b, T_c under the homothety with center P and coefficient 2. Thus we immediately see that the points P_a, P_b, P_c are collinear. It remains to be shown that the orthocenter H lies on the same straight line.

In the diagram, T_aT_b is the simson line ΔABC. To prove the assertion, we shall show that HP_b||T_aT_b.

First, since PT_a is perpendicular to BC and PT_b is perpendicular to AC, the quadrilateral PT_aT_bC is cyclic. The inscribed angles T_aT_bP and T_aCP stand on the same arc T_aP and are, therefore equal:

(1)	∠T_aT_bP = ∠T_aCP.

HP_b is the reflection of K_bP in AC, while HK_b is perpendicular to AC. Therefore,

(2)	∠BK_bP = ∠K_bHP_b.

Angles BK_bP and BCP (= T_aCP) are equal as inscribed into the same circle and being subtended by the same arc BP:

(3)	∠BK_bP = ∠BCP.

The combination of (1)-(3) gives a crucial for the proof identity:

(4)	∠T_aT_bP = ∠P_bHH_b.

Since one pair of the sides of these angles (PT_b and HH_b are both perpendicular to AC) are parallel, the same holds for the other pair of the sides: HP_b||T_aT_b.

Corollary

The simson line of a point P on the circumcircle of a triangle bisects the segment PH joining the point to the orthocenter H.

References

H.S.M. Coxeter, S.L. Greitzer, Geometry Revisited, MAA, 1967
R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995.

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	Circumcircle of Three Parabola Tangents
	Angle Bisector in Parallelogram
	Simsons and 9-Point Circles in Cyclic Quadrilateral
	Simsons of Diametrically Opposite Points
	Simson Line From Isogonal Perspective
	Pentagon in a Semicircle
	Simson Line in Disguise
	Two Simsons in a Triangle
	Carnot's Theorem
	A Generalization of Simson Line

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Reflections of a Point on the CircumcircleWhat Is This About? A Mathematical Droodle

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.