Geometry

All Triangles are Isosceles!

1. Start with a random triangle 4ABC.

2. Locate it’s mid-point D. (Prop 10)

3. Create a perpendicular line through AB, going through D. (Prop 11)

4. Construct a line bisecting ∠ACB and locate the intersection between this line and the previously drawn perpendicular to AB going through D. Call this point E. (Prop 9)

5. Construct perpendicular line segments to the two sides AC and BC through E. That is, construct line segments EF and EG that are perpendicular to the corresponding sides of 4ABC. (Prop 12)

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6. Connect EA and EB (i.e. draw those line segments).

Now something whacky is about to happen!

Let’s start by looking at 4CFE and 4CGE.

They share a side, and have two angles equal. So by AAS (Prop 26), they are congruent. And so

CF = CG (1)

Also, EF = EG (2)

Now consider 4EAD and 4EBD.

These two have ED in common, and by construction, AD = BD. So by SAS (Prop 4), they are congruent. And so

AE = BE (3)

Finally, look at 4AFE and 4BGE.

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These are both right angled triangles with two sides equal (equation 2 and equation 3). So by RASS (proved today in class!) they are congruent. And so

FA = GB (4)

So now we have:

CF = CG (Equation 1)

FA = GB (Equation 4)

CF + FA = CG + GB (Adding equations 1 and 4)

CA = CB

Thus the “random triangle” we started with, is actually isosceles! So all triangles are isosceles. And actually by the same argument on different sides, equilateral! YIKES!!!

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