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My Kingdom for an Auxillary Line Segment

Note A: This conjecture sounds simple...until you try it. But, if you can introduce the needed additional "auxillary line segment," the proof almost falls out on its own. And, once it does, try to prove the Corallary as well.

Conjecture: If a triangle is inscribed in a circle, the product of any two sides equals the product of the altitude on the third side by the diameter.

Note: In the drawing, the claim is that (CA)(CB) = (CD)(CE).


Corallary: The area of any triangle inscribed in a circle is the product of its three sides divided by twice the diameter.

Note: In the drawing, the claim is that the area of triangle ABC = [(CA)(CB)(AB)]/[(2)(CE)].

 

Source: J. Thompson. Geometry for the Practical Man, 1934. pp. 188, 248-249


Hint: The needed "auxillary line" is one of these three line segments: AE, DE, or BE.

You need to play with each to see which is the most useful...actually, two of those will work equally well!

 


Solution Commentary: Suppose segment BE is introduced. Then, consider these "falling out " ideas:

  • Triangles CBE and CDA are right triangles...why?
  • Angles CAB and CEB are congruent...why?
  • Triangles CBE and CDA are similar....why?
  • Therefore, their corresponding sides are proportional...that is... ?
  • Cross-multiply and you are done!

As for the Corallary, you have the established product...multiply both sides of product by BC. Also, you know area of triangle ABC = (1/2)(AB)(CD). Combining these two expressions should give you the necessary relationship.