A regular hexagon, two squares and a semicircle connecting two vertices. Is the red vertex located on the semicircle?

Scroll down for a solution to this problem.

## Solution

Yes, the vertex is on the semicircle, because the angle in the semicircle is 15°+75°=90°.

## Poem

The dance

Dancing day and night

Having fun and glide

With a dancer, my inspiration

He is my guide and vision

We perform every hour

With some friends on the floor

Moving with our drums

Dancing in our own rhythm

Stepping and twirling

Letting the fire burn

To inspire our souls

## One reply on “The dance”

Let the hexagon be ABCDEF clockwise with AB top horizontal side and the two squares be EDGH and FEIJ.

Let the side of hexagon be 2. Join AJ, JH and JE

In triangle AJE, AE=2√3, JE = √8 and angle AEJ = 45+30=75. Using cosine rule, AJ^2= 14.9282.

In triangle JEH, EH=2, JE = √8 and angle JEH = 60+45=105. Using cosine rule, JH^2 = 14.9282.

Also AH^2 = (AE + EH)^2 = (2*√3 + 2)^2 = 29.8564.

From triangle AJH, using above three values, the triangle is found to be a right angled triangle with angle AJH = 90°

Hence the point J must be on the circumference of the semicircle shown.