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Intermediate

The owl of Athena

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.

Solution by Asmit Dey.

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

Bella

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