Proof of the Simpson Line

By: Ginger Rhodes

 

 

Given: DABC is inscribed in circle O, The Pedal point P is on the circle O, PS^AC, PT^AB, and PR^BC [These are segments]

 

Prove: Points R, S, and T, the Simpson line, are collinear

 

 

Proof:

 

<PSA and <PTA are both right angles, and therefore supplementary. So <SPT and <TAS are supplementary and opposite angles in Quadrilateral PSAT are supplementary.

 

Now, <PST and <PAT have the same intercepted arc, which means the angles are congruent. Similary, points P, R, C, and S are cyclic.

 

 

So, <PCR and <PSR have the same intercepted arc, which means the angles are congruent.

 

By hiding some of the lines it is easier to see DPAB@DPCB [same intercepted arcs].

 

 

 

So, <PST congruent <PAT, <PCR congruent <PSR, and <PAB congruent <PCB.

 

Notice, <PAB is another name for <PAT and <PCB is another name for <PCR.

 

Using the transitive property we can establish <PST @ <PSR, which implies S, T, and R are collinear!

 

RETURN