# Proving isosceles using centroid and medians

• MHB
• slwarrior64
In summary, the conversation discusses using the centroid property and Stewart's Theorem to write an equation involving the three sides of a triangle. After simplifying, it is found that the only possible solution is when $b=c$ or $AB=AC$. No synthetic proof is seen for this equation.
slwarrior64
I can definitely do this in the opposite direction, but

Letting $AB=c$, $AC=b$ and noting that the usual property of the centroid tells us that $GC = \frac{2}{3}$ of the median from $C$ and $GB = \frac{2}{3}$ of the median from $B$ we can use Stewart's Theorem to write everything in terms of the three sides $a,b,c$. It is an involved equation requiring two squarings to get rid of all radicals. I used Wolfram Alpha to double check my computations. After simplifying everything I end up with

$(b-c)^2 \left[ (b-c)^2-a^2 \right] =0$.

The bracket fails the triangle inequality which only leaves $b=c$ or $AB=AC$. I can't see any synthetic proof.

## 1. What is the definition of an isosceles triangle?

An isosceles triangle is a triangle with two sides of equal length and two angles of equal measure. This means that the base angles (the angles opposite the equal sides) are congruent.

## 2. How can the centroid and medians be used to prove that a triangle is isosceles?

The centroid of a triangle is the point where all three medians intersect. A median is a line segment drawn from a vertex to the midpoint of the opposite side. In an isosceles triangle, the medians from the equal sides will be equal in length, and they will intersect at the centroid. Therefore, if we can show that the medians are equal and intersect at the centroid, we can prove that the triangle is isosceles.

## 3. What is the formula for finding the centroid of a triangle?

The formula for finding the centroid of a triangle is (x,y), where x is the average of the x-coordinates of the three vertices and y is the average of the y-coordinates of the three vertices. In other words, the centroid is the point of intersection of the three medians.

## 4. Can a triangle be isosceles if it does not have a centroid?

No, a triangle cannot be isosceles if it does not have a centroid. The centroid is a defining feature of an isosceles triangle, as it is the point where the medians intersect. If a triangle does not have a centroid, it cannot have medians, and therefore cannot be isosceles.

## 5. Are there other ways to prove that a triangle is isosceles?

Yes, there are other ways to prove that a triangle is isosceles. Some other methods include using the congruence of sides and angles, using the Pythagorean theorem, or using the properties of parallel lines and transversals. However, using the centroid and medians is a common and efficient method for proving isosceles triangles.

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