A Fundamental Theorem of Similarity
If F and Fâ² are any two directly similar1 figures with the vertices P in F corresponding to vertices Pâ² in Fâ², and the lines PPâ² are divided in the ratio of r : 1âr, that is, at points Pâ²â² = (1 â r)P + rPâ², then the new figure Fâ²â² formed by the points Pâ²â² is directly similar to F and Fâ².
Illustration: The dynamic figure below illustrates the theorem for two directly similar quadrilaterals IJKL and Iâ²Jâ²Kâ²Lâ². Drag any of the vertices of these two directly similar quadrilaterals, or B to vary the ratio r above, or E or H to change the position or orientation of Iâ²Jâ²Kâ²Lâ². Click on the 'Show Ratio Measurements' button to display the ratios of corresponding sides of Fâ²â² and Fâ².
(Note1: Two similar figures are 'directly similar' if their corresponding angles have the same rotational sense (and are not reversed in relation to each other as in a reflection).
Challenge
Can you explain why (prove that) the theorem is true?
A Fundamental Theorem of Similarity
References
This theorem can be proved in various ways. Here are some references (in chronological order) for the reader to consult:
1) DeTemple, D., & Harold, S. (1996). A round-up of square problems. Mathematics Magazine, 69(1), 15â27. https://doi.org/10.1080/0025570X.1996.11996375
2) De Villiers, M. (1998). Dual generalisations of Van Aubelâs theorem. The Mathematical Gazette, 82, 405â412.
3) Abel, Z. R. (2007). Mean geometry. (Free download).
4) Fried, M. (2021). From any two directly similar figures, produce a new one. International Journal of Geometry, 10(3), 90â94. (Free download).
Some Applications
This similarity theorem is very useful and can be applied to many interesting problems. For example, view the following dynamic geometry sketches & papers:
1) Van Aubel's Theorem and some Generalizations
2) An associated result of the Van Aubel configuration and some generalizations
3) Associated Van Aubel: Different Similar Quadrilateral Arrangements
4) Associated Van Aubel: Similar Triangle Arrangements
5) My 2023 paper An associated result of the Van Aubel configuration and its generalization in the Int. Journal of Math Ed in Sci & Technol. extensively explores & uses this theorem for a wide range of results (as shown at Links 2-3 above).
6) Zachary Abel's 2007 paper Mean geometry gives some neat examples of mathematical olympiad problems to which the theorem readily applies.
7) applies to Dao Than Oaiâs hexagon generalization of Napoleonâs theorem and gives a further generalization.
Related Links
FinslerâHadwiger theorem plus Gamow-Bottema's Invariant point
An associated result of the Van Aubel configuration and some generalizations
Dao Than Oaiâs hexagon generalization of Napoleonâs theorem
Associated Van Aubel: Different Similar Quadrilateral Arrangements
Associated Van Aubel: Similar Triangle Arrangements
Van Aubel's Theorem and some Generalizations
Van Aubel Centroid & its Generalization
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
Some Variations of Vecten configurations
Napoleon's Theorem: Generalizations & Converses
*********
Free Download of Geometer's Sketchpad
Copyright © 2019 KCP Technologies, a McGraw-Hill Education Company. All rights reserved.
Release: 2020Q3, semantic Version: 4.8.0, Build Number: 1070, Build Stamp: 8126303e6615/20200924134412
Back to "Dynamic Geometry Sketches"
Back to "Student Explorations"
Created 9 September 2022 by Michael de Villiers, using WebSketchpad; updated 20 Nov 2023; 13 August 2025.
RetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4