**Exploration**

**Construction**: Follow the step by step construction described below:

Step 1: Construct a dynamic △*ABC* and a dynamic line segment *QR*.

Step 2: Measure the ratios *AB/BC* and *AC/BC*.

Step 3: Mark ratio *AB/BC* as a ‘Scale Factor’, and mark *Q* as a ‘Center’, then ‘Dilate’ the point *R* from *Q* as centre with the marked ratio *AB/BC*. With *Q* as centre draw a circle with radius *QR'* where *R'* is the image of the preceding dilation.

Step 4: Mark ratio *AC/BC* as a ‘Scale Factor’, and mark *R* as a ‘Center’, then ‘Dilate’ the point *Q* from *R* as centre with the marked ratio *AC/BC*. With *R* as centre draw a circle with radius *Q'R* where *Q'* is the image of the preceding dilation.

Step 5: Construct the intersection of the two circles, and label one of the intersections as *P*. (We ignore the other intersection, say *P'*, since △*P'QR* is congruent to △*PQR*).

**Explore**: In the interactive sketch below, Steps 1-2 have already been done. Click on the 'Show Steps 3-5' buttons to display the next steps.

1) What do you notice about triangles *ABC* and *PQR*? Check your observation by dragging any of the red points.

2) Can you formulate your findings as a theorem?

Forgotten Similarity Theorem

**A Forgotten Similarity Theorem**

If for two triangles *ABC* and *PQR* any two pairs of ratios hold from the following three pairs of corresponding ratios, *AB/BC = PQ/QR, BC/AC = QR/PR* and *AB/AC = PQ/PR*, then △*ABC* is similar to △*PQR*.

**Challenge**

Can you logically explain why (prove that) the above theorem is true?

**Note**

Note that the theorem above is equivalent to the SSS-similarity theorem - it implies the SSS-similarity theorem and is implied by it. Perhaps that is the reason why it is not mentioned in geometry texts?

**Reference**

A paper A Forgotten Theorem for Triangle Similarity? with a proof of the above result has been published in the *Learning & Teaching Mathematics (LTM)* journal, Dec 2022, no. 33, published by the Association for Mathematics Education of South Africa.

Copyright © 2019 KCP Technologies, a McGraw-Hill Education Company. All rights reserved.

Release: 2020Q2, Semantic Version: 4.6.2, Build Number: 1047, Build Stamp: 139b185f240a/20200428221100

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Created 20 August 2022 by Michael de Villiers, using *WebSketchpad*; updated 19 Jan 2023.