A triangle with a median and another cevian. Two areas are given. What’s the total area?
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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.
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
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With her owl symbol for philosophers.