Heights and Distances
Mathematics – Advanced University Level (MIT‑style)
Manipal MET – Mathematics • “Heights and Distances”
Goal – Build a rigorous, first‑principles understanding of heights (altitudes) in planar geometry and distances between arbitrary geometric objects in Euclidean space.
Audience – Upper‑division undergraduate students with a solid background in geometry, linear algebra, and basic calculus.
1. Overview
- Heights (or altitudes) are perpendiculars dropped from a vertex to the opposite side (or its extension) in a triangle.
- Distances measure how far apart two points, a point and a line, two lines, a point and a plane, etc., are in Euclidean space.
- Both concepts are fundamental in analytic geometry, optimization, physics, engineering, and computer graphics.
2. Basic Definitions
| Symbol | Meaning | Remark |
|---|---|---|
| ( \triangle ABC ) | Triangle with vertices (A,B,C) | Assume (A,B,C) not collinear |
| ( h_a ) | Altitude from (A) to side (BC) | May fall outside (BC) if (\triangle ABC) is obtuse at (A) |
| ( \overline{AB} ) | Line segment from (A) to (B) | Length ( |
| ( \ell ) | Line (affine) | Infinite in both directions |
| ( \Pi ) | Plane | Infinite 2‑D subspace |
| ( \text{dist}(X,Y) ) | Euclidean distance between points (X) and (Y) | ( \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+\dots} ) |
3. Heights (Altitudes) in a Triangle
3.1 Construction
For vertex (A), let (D) be the foot of the perpendicular from (A) onto the line (BC).
The segment (AD) is the altitude (h_a).
3.2 Concurrency of Altitudes
Theorem (Orthocenter Concurrency).
In any triangle ( \triangle ABC ), the three altitudes (AD, BE, CF) are concurrent at a single point (H) called the orthocenter.
Proof Sketch (using Ceva’s theorem):
Let (D, E, F) be the feet on (BC, CA, AB) respectively.
Define ratios
[
\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1.
]
Ceva’s theorem states that the cevians (AD, BE, CF) concur iff the above product equals 1.
By dropping perpendiculars, one shows each ratio equals (\frac{\sin \angle B}{\sin \angle C}), etc., and the product indeed equals 1. ∎
3.3 Length of an Altitude
Let (\Delta = \text{area}(\triangle ABC)).
By the area formula ( \Delta = \frac{1}{2} \cdot \text{base} \cdot \text{height} ) applied to side (BC):
[ h_a = \frac{2\Delta}{|BC|}. ]
If the coordinates of (A,B,C) are known, (\Delta) can be computed via the cross product: [ \Delta = \frac{1}{2}|(B-A)\times(C-A)|. ]
Edge Case – Obtuse Triangle:
If (\angle A > 90^\circ), the foot (D) lies on the extension of (BC) beyond (C). The altitude length (h_a) is still given by the same formula; the sign of (h_a) (if we keep orientation) indicates whether the foot lies inside or outside the segment.
4. Distances in Euclidean Space
4.1 Point‑to‑Point
For