3
votes
Orthocenter: The "Bad Boy" of Distinguished Points in a Triangle
I don't know if you will like this proof, which should IMHO be considered as a coordinate-free one ...
For all point $M$ in the plane :
$$\begin{eqnarray}\overrightarrow{AM}.\overrightarrow{BC}+\...
- 7,682
3
votes
Largest Area Triangle in the Vesica Piscis
Here is a solution in the "17th century spirit" where extremal solutions were found based on the computation of infinitesimal quantities.
I assume that we look for an optimal solution under ...
- 83.9k
2
votes
Accepted
Orthocenter: The "Bad Boy" of Distinguished Points in a Triangle
Let $BD$ and $CE$ be the altitudes that meet at $H$. We'll show that $AH$ is an altitude too, which will prove that the three altitudes in a triangle are concurrent.
Let $AH$ intersect $BC$ at $K$.
...
- 1,275
1
vote
2-D scalene obtuse triangle trigonometry.
For the purposes of this answer, I am going to label the length of $AD$ as $d$, the length of $CD$ as $d$ (they are equal in the question), angle $ADC$ as $x$ and angle $BDC$ as $y$.
By the cosine ...
- 2,792
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