Can You Prove This Identity? 1/a + 1/b + 1/c = 1/d

Added: 2026-05-17

[00:00:00]Can you prove this elegant identity? 1 over a plus 1 over b plus 1 over c equals 1 over d. Let's explore it using these two nested squares. Method one, geometric similarity. Based on the similarity between triangles ABC and BGH, we can determine the length of segment Y. Similarly, by comparing triangle ABD with the triangle AEF, we can determine the length of segment X.

[00:00:26]Notice that the entire base has length C. So, the pieces must add together. X plus D plus Y equals C. Divide both sides by DC, simplify the fractions, and the identity appears beautifully. Method two, coordinate geometry. Place the figure on the XY plane and write equations for the two diagonal lines.

[00:00:45]Next, find where those lines intersect the horizontal line at height D. That gives us the length of the top edge of the inner square. After a little algebraic rearranging, we arrive at the exact same identity. Two completely different methods, one elegant truth.