A point equidistant from the vertices or the sides of a polygon
If M is equally far from the vertices of a triangle, its projection is the circumcentre; if equally far from the sides, the incentre. Find the distance from M to the plane.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- Ukraine: grade 10
Guide
From the Ukrainian Grade 10 textbook (algebra and calculus with geometry), §5, item 35 'Perpendicular and slant': the key problem about an equidistant point.
If a point M is equally far from all vertices of a polygon, its projection onto the polygon's plane is the centre of the circumscribed circle: equal slants MA = MB = MC have equal projections OA = OB = OC.
If M is equally far from the lines containing the sides, its projection is the centre of the inscribed circle (by the theorem of three perpendiculars).
Then work in the right triangle MOA, where MO is the perpendicular, MA the slant and OA the projection: MO = √(MA2 − OA2).