Assignment #8 ~ That Triangle is, ah, Cute!
Investigating Ratios
Given an Acute Triangle and its Circumcircle!
By Kevin Mylod

In this investigation, we are to construct acute triangle ABC (in blue above), its orthocenter H (in red), and its circumcircle C1 (in pink). To make your own construction, link on to the GSP script.

If we construct the three altitudes of the triangle and extend them to intersect with circumcircle C1, we can then construct segments AD, AP, BE, BQ, CF, and CR, as shown above.

Looking at lengths of these segments carefully, there might be some relationship between the lengths of AD and AP, BE and BQ, and CF and CR. In fact, the relationship is within their ratios! What happens if we add the three ratios together? Does this value change as we move our three given points (A, B, or C) around or does the value remain constant?

If you went ahead and made your construction from the linked script, continue by measuring the lengths in GSP and finding the three ratios mentioned above. Or, you can click onto the figure below to see the GSP construction with the completed ratio calculations.

By following the instructions, you notice that the sum of the three ratios remains constant at 4 units. The obvious question you are asking yourself is why? The answer to this question is found in the proof below.

Proof:

Q.E.D.

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