## A Fundamental Theorem of Similarity

A Fundamental Theorem of Similarity
If F and F′ are any two directly similar1 figures with the vertices P in F corresponding to vertices P′ in F′, and the lines PP′ are divided in the ratio of r : 1−r, that is, at points P′′ = (1 − r)P + rP′, then the new figure F′′ formed by the points P′′ is directly similar to F and F′.
Illustration: The dynamic figure below illustrates the theorem for two directly similar quadrilaterals IJKL and I′J′K′L′. 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 I′J′K′L′. Click on the 'Show Ratio Measurements' button to display the ratios of corresponding sides of F′′ and F′.
(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?

#### Web Sketchpad .sketch_canvas { border: medium solid lightgray; display: inline-block; } 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), 15–27. https://doi.org/10.1080/0025570X.1996.11996375
2) De Villiers, M. (1998). Dual generalisations of Van Aubel’s theorem. The Mathematical Gazette, 82, 405–412. https://doi.org/10.2307/3619886
4) Fried, M. (2021). From any two directly similar figures, produce a new one. International Journal of Geometry, 10(3), 90–94. (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 Feb 2022 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. (If free download copies are no longer available, contact me directly).
6) 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 Oai’s hexagon generalization of Napoleon’s theorem and gives a further generalization.