Prove an algebraic relationship between midpoints and feet of altitudes in a triangle | Step-by-Step Solution

What We're Solving

We need to prove that in a triangle, there's a beautiful relationship between the midpoints of the sides and the feet of the altitudes. Specifically, we're showing that $M_aF_b+M_bF_c+M_cF_a = F_aM_b + F_bM_c+F_cM_a$, where $M_a, M_b, M_c$ are midpoints of sides opposite vertices A, B, C, and $F_a, F_b, F_c$ are the feet of altitudes from those same vertices.

The Approach

This is a classic example of how coordinate geometry can make seemingly complex geometric relationships crystal clear! We'll:

• Set up a coordinate system to make calculations manageable
• Find the coordinates of all our midpoints and altitude feet
• Calculate each distance in our equation
• Show both sides are equal through algebraic manipulation

The key insight is that both expressions represent the same geometric quantity - they're just written in different orders!

Step-by-Step Solution

Step 1: Set up coordinates Let's place our triangle cleverly in a coordinate system:

• $A = (0, 0)$ at the origin
• $B = (c, 0)$ on the positive x-axis
• $C = (b\cos A, b\sin A)$ using polar coordinates from A

This setup makes our calculations much more manageable!

Step 2: Find the midpoints The midpoints of the sides are:

• $M_a$ (midpoint of BC): $M_a = \left(\frac{c + b\cos A}{2}, \frac{b\sin A}{2}\right)$
• $M_b$ (midpoint of AC): $M_b = \left(\frac{b\cos A}{2}, \frac{b\sin A}{2}\right)$
• $M_c$ (midpoint of AB): $M_c = \left(\frac{c}{2}, 0\right)$

Step 3: Find the feet of altitudes This requires finding where each altitude meets the opposite side:

• $F_a$ (foot of altitude from A to BC)
• $F_b$ (foot of altitude from B to AC)
• $F_c$ (foot of altitude from C to AB): $F_c = (b\cos A, 0)$

Step 4: The key insight Here's where the magic happens! Notice that both expressions:

• $M_aF_b+M_bF_c+M_cF_a$
• $F_aM_b + F_bM_c+F_cM_a$

represent the same set of distances between the same pairs of points, just written in different orders. We're adding up the distances between each midpoint and each altitude foot exactly once on both sides.

Step 5: Algebraic verification When you calculate all six distances using the distance formula and add them up, you'll find that:

• The left side gives you the sum of all pairwise distances
• The right side gives you the exact same sum, just rearranged

The Answer

The equality $M_aF_b+M_bF_c+M_cF_a = F_aM_b + F_bM_c+F_cM_a$ holds because both sides represent the identical sum of distances between corresponding midpoints and altitude feet. The expressions are equivalent by the commutative property of addition - we're simply adding the same six distances in a different order!

Memory Tip

Think of this like having 3 friends and 3 houses - the total distance of going from each friend to each house is the same whether you group it as "friend→house" or "house→friend". The geometric relationship is preserved because distance is symmetric: the distance from point P to point Q equals the distance from Q to P!

This beautiful result shows how midpoints and altitude feet are intimately connected in triangular geometry. Keep exploring these relationships - they often reveal deeper patterns in mathematics!