A triangle with a median and another cevian. Two areas are given. What’s the total area?

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## Solution

The total area is 40.

First connect points E and C and suppose the centre triangle has area x. From the congruence of BE and ED it follows that triangle ECD must have area 5+x. From the same congruence we infer the area BFA as 9+2x.

Now because their heights are the same, the triangle areas EFC : FCA = EFB : BFA. So we get x/9=5/(9+2x). This can be rewritten as a quadratic equation with only one positive root: x=3. Adding all areas gives a total of 40.

Notice we never used the right angles in this solution, so the problem is over-constrained.

## Poem

Goddess of wisdom and power Athena

With her wide eyes owl

Fighting giants near Mount Etna

The owl was her friend flying in the sky

Protecting her with strong cries

Beloved from Gods and soldiers

Teaching how to live together

With her owl symbol for philosophers.

Divine Athena