Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
998 views
in Co-ordinate geometry by (63.5k points)

Find a proof that the three altitudes of a triangle are concurrent using classical Euclidean geometry.

1 Answer

+1 vote
by (64.8k points)
selected by
 
Best answer

Let AD be the altitude perpendicular to BC, let BE be the altitude perpendicular to AC and let AD meet BE at H. Finally let C H meet AB at F. We must prove FC is perpendicular to AB.

Join DE. Let α = ∠BC F. Then ECDH is a cyclic quadrilateral (supplementary opposite angles), so ∠DE H = α (angles in the same segment).

Similarly, AEBD is a cyclic quadrilateral and ∠D AB = α.

From 4ABD, ∠ABD = 180 −90 −α = 90 −α.

Finally, in 4CBF, ∠BFC + α + 90 − α = 180, so ∠BFC = 90, as required.

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

...