Parallelogram and Similar Triangles
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A Mathematical Droodle

9 October 2015, Created with GeoGebra


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Copyright © 1996-2017 Alexander Bogomolny

The applet is intended to suggest the following proposition:

If ABCD is a parallelogram and P, Q (positioned as shown in the applet) are such that triangles PAB and BCQ are similar (with P corresponding to B, A to C, and B to Q), then each is similar to triangle PDQ.

Note that the statement is a double generalization of a problem by V. Thébault. For one, the square is replaced by a parallelogram. Also, two equilateral triangles are replaced by a pair of arbitrarily shaped similar triangles.

Parallelogram and three similar triangles

Proof 1

To say that "triangles PAB and BCQ are similar" means precisely that

∠PAB = ∠BCQ = α say,
∠ABP = ∠CQB and
∠BPA = ∠QBC.

Moreover, we know also that PA/BC = AB/CQ (since the two triangles are automatically proportional). In the parallelogram we know that BC = AD and AB = DC, so that we have

PA / AD = DC / CQ.

Since opposite angles in a parallelogram are equal we have ∠BAD = ∠BCD = β say, so

∠DAP = ∠DCQ = α + β.

Hence, using SAS version of similarity, triangles PAD and DCQ are similar, so ∠APD = ∠CDQ = γ say, and ∠ADP = ∠CQD = δ say. From triangle DCQ we have

α + β + γ + δ = 180°

(angles in a triangle). Since ∠ADC + ∠BCD = 180° (adjacent angles in a parallelogram), we conclude that

∠PDQ = α.

Since triangles PAD and DCQ are similar, we have

PA / PD = DC / DQ = AB / DQ,

so PA / AB = PD / DQ. Finally, since ∠PAB = ∠PDQ = α, we conclude that triangles PAB and PDQ are similar (SAS version of similarity.)

Proof 2

Let K be the intersection of the diagonals in the parallelogram and L and M the midpoints of BP and BQ respectively. Then triangle KLM is similar to triangles APB and CBQ by the Three Similar Triangles Theorem. Triangles KLM and DPQ are similar as having pairwise parallel sides.

Proof 3

William McWorter has noticed that the linear combinations in Proof 2 can be replaced with common vector (or affine) sums. Taking B as the origin, we have D = A + C, P = P + B, and Q = B + Q, which makes ΔDPQ similar to ΔAPB and ΔCBQ directly, without the intermediate ΔKLM.


  1. A. D. Gardiner, C. J. Bradley, Plane Euclidean Geometry: Theory and Problems, UKMT, 2005

Related material

  • Angle Bisectors In Rectangle
  • Segment Trisection Induced by Parallels to Medians
  • The Nature of Pi
  • Two Triples of Similar Triangles
  • Three Similar Triangles
  • Similar Triangles on Sides and Diagonals of a Quadrilateral
  • Point on Bisector in Right Angle
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    Copyright © 1996-2017 Alexander Bogomolny


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