# Every triangle is equilateral

in mathematics •  2 years ago

# Every triangle is equilateral

Here is a small proof of the equality "2=1"

• Choose two numbers a and b such that a=b
• Multiply by a to get : a²=ab
• Substrating gives : a²-b²=ab-b²
• Factoring by a-b gives : (a-b)(a+b)=(a-b)b
• Divide by (a-b) to get a+b=b
• Finally, choose a=b=1 to get 2=1

In this "proof", the trick is not really well hidden : if a=b=1, then we shouldn't be allowed to divide by a-b as it would be zero.

In this post I'll be presenting a "proof" of the obviously fake assertion that every triangle is equilateral. The flaw in this proof is not easy to find and I'll leave it as a challenge to find it, or give it on request in the comments.

Angle bisector and segment bisector

Take a triangle ABC and draw the angle bisector of the angle in B, and the segment bisector of [AC]. They intersect at a point named E.  Project the point E on the two segments [AB] and [BC], and get two new points, H and I The reasoning is the following :

• D2 being an angle bisector, and E being on D2, the lenghts BI and BH are equals, as well as the lenghts EH and EI
• D1 being a segment bisector, the lenght EA and EC are equals.
• The triangles AEH and ECI have two pairs of equal sides (EI=EH and EC=EA) and the angles EHA and EIC are equal because they are both right angles. Thus, the triangles AEH and ECI are equal, and it gives the equality HA=IC
• We have proven that BH=BI and that HA=IC, and the sum of these equalities gives BA=BC, proving that ABC is an isocele triangle. Repeat the proof on one other side to conclude that the three sides are equals and that ABC is an equilateral triangle.