Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade None
Question:
If in a triangle $ABC$, atleast one of the following points, orthocentre, centroid, incentre and circumcentre lie outside the triangle, then :
Triangle is obtuse angled
Exactly 2 of these centres will lie outside the triangle
Incentre may be collinear with other three centres
Atleast one of the ex-radii is smaller than the inradius of the triangle
Step-by-Step Solution
Key Concept: The relative positions and magnitudes of the four triangle centers depend on whether the triangle is acute, right, or obtuse.
In an obtuse triangle, the circumcenter and orthocenter lie outside the triangle. If the triangle is obtuse, the incenter may be collinear with the other three centers. Using the formulas $r = \frac{\Delta}{s}$, $r_1 = \frac{\Delta}{s-a}$, $r_2 = \frac{\Delta}{s-b}$, $r_3 = \frac{\Delta}{s-c}$, we have $r < r_2$ and $r < r_3$ for an obtuse triangle.
Correct Answer: 1,2,3