Let be an acute angled triangle and point chosen differently from . Prove that is the orthocenter of triangle if and only if
Solution of my students.
If is the orthocenter of triangle then by a well-known theorem, we have By triangle is acute: therefore , so by we have is proved.
If is true then by , in interior of the triangle . By a well-known theorem, we have
Use and we have
, hence . Similary, . Now, easy see that is orthocenter of the triangle .
Note: If and then .