给定可组成拼接三角形的线段集合，求解最大距离

Hey there, let's break this down step by step—I’ve been in similar spots when dealing with complex triangle formations from connected segments, so I get exactly where you’re stuck. Here’s a practical, actionable approach to calculate heights for those new triangles:

The core idea is that regardless of how segments are joined to form new triangles, you can always rely on basic triangle properties and coordinate geometry (if you have that data) to find the height. Here’s how to tackle it:

1. First, Pin Down the New Triangle’s Core Details #

• Start by identifying the three vertices of the new triangle. When segments are added and spliced with the original triangle, every new triangle will have three distinct vertices. If you don’t have their coordinates, calculate them using the lengths of the segments and the original triangle’s known vertices.
• If you only have segment lengths (no coordinates), confirm which side you want to use as the base, and note which vertex is opposite that base (this is the vertex from which you’ll measure the height).

2. Use Area as a Universal Bridge to Find Height #

This trick works for any triangle, no matter how it’s formed:

• First, calculate the area of the new triangle. If you know all three side lengths, use Heron’s Formula:
  Area = √[s(s-a)(s-b)(s-c)]
  where s = (a + b + c)/2 (a, b, c are the lengths of the triangle’s three sides)
• Once you have the area, pick your chosen base length, then rearrange the area formula to solve for height:
  Height = (2 * Area) / base

3. Coordinate-Based Method (If You Have Vertex Coordinates) #

If you have (x,y) coordinates for the new triangle’s three vertices (e.g., A(x₁,y₁), B(x₂,y₂), C(x₃,y₃)):

• Calculate the length of your chosen base (say, AB) using the distance formula:
  base = √[(x₂ - x₁)² + (y₂ - y₁)²]
• Compute the area using the Shoelace Formula:
  Area = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
• Again, use Height = (2 * Area) / base to get the height from the opposite vertex (C) to the base AB.

4. Leverage Shared Edges for Connected Triangles #

When the new triangle shares a side with the original one, you can use that shared edge to simplify calculations:

• Treat the shared edge as the base for both triangles. If you already know the original triangle’s height relative to this base, you can often use the position of the new vertex (from the added segments) to find the new height. For example, if the new vertex sits along a line parallel to the shared base, the height might be additive or subtractive depending on how the triangles are arranged.

Pro Tip: Sketching a rough diagram of the shape always helps. Visualizing which vertices form the new triangle and which side you’re measuring the height against makes it way easier to plug the right values into formulas.

内容的提问来源于stack exchange，提问作者Mert AKEL