Assignment 8

Investigation 11


Construct any acute triangle ABC and its circumcircle.  Construct the three altitudes AD, BE, and CF.  Extend each altitude to its intersection with the circumcircle at corresponding points P, Q, and R.






Proof of Claim:


First break the segments into ratios as follows:



Next we add:




to get:



First, consider the first ratio,




HD is the altitude of the triangle BHC and AD is the altitude of the triangle ABC, so we can use them in the formula for the areas of the triangles BHC and ABC:







So, plugging in our original equation gives us:



Note :



Since, those three triangles make up the triangle ABC:

And since




We get that