Altitudes and the Power of a Point
What Is This About?
A Mathematical Droodle

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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

The applet provides an illustration to a problem from an outstanding collection by T. Andreescu and R. Gelca:

Let ABCD be a convex quadrilateral inscribed into a semicircle s of diameter AB. The lines AC and BD intersect at E and the lines AD and BC at F. The line EF intersects semicircle s at G and the line AB at H. Prove that E is the midpoint of the line segment GH if and only if G is the midpoint of the line segment FH.

power of a point and altitudes

The statement follows from a more general fact that GH is the geometric mean of EH and FH:

(1) GH2 = EH·FH;

which, for example, also implies that if E divides GH in the golden ratio then the same is true of G with respect to FH. So, let's prove (1).

The right triangles AFH and ABD share an angle at A and are thus similar. Similarly, the right triangles ABD and EBH that share an angle at B are also similar. By transitivity, triangles AFH and EBH are similar. The latter similarity implies the proportion:

AH/EH = FH/BH

which is to say

(2) AH·BH = EH·FH.

But ABG is a right triangle with hypotenuse GH so that AH·BH = GH2 which together with (2) implies (1).

Note that AH·BH is the power of point H with respect to the given circle.

References

  1. T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, 5th printing, 1.3.9 (p. 12)

Power of a Point wrt a Circle

  1. Power of a Point Theorem
  2. A Neglected Pythagorean-Like Formula
  3. Collinearity with the Orthocenter
  4. Circles On Cevians
  5. Collinearity via Concyclicity
  6. Altitudes and the Power of a Point
  7. Three Points Casey's Theorem
  8. Terquem's Theorem
  9. Intersecting Chords Theorem
  10. Intersecting Chords Theorem - a Visual Proof
  11. Intersecting Chords Theorem - Hubert Shutrick's PWW

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Copyright © 1996-2017 Alexander Bogomolny

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