I don’t understand these concepts. Can you break them down into simple and detailed explanations so that they are easy to understand?

Certainly! The term “公垂线” (gōng chuíxiàn) is a concept from geometry, particularly in the context of triangles and lines. It can be translated to “concurrent perpendiculars” or “height lines” in English. Let's break it down step by step: ### What is a "Perpendicular Line"? 1. **Definition**: A perpendicular line is a line that meets another line at a right angle (90 degrees). 2. **Example**: If you imagine a straight line on a piece of paper and you draw another line that crosses it making a "T" shape, then the two lines are perpendicular to each other. ### Understanding "公垂线" in Geometry 1. **In Triangles**: In the context of triangles, each vertex (corner point) can have a line drawn from it to the opposite side that is perpendicular to that side. These lines are called the "heights" or "altitudes" of the triangle. - **Example**: If you have triangle ABC, you can draw a perpendicular line from point A to side BC. This line represents the height from A to the base BC. 2. **"Concurrent" Concept**: The lines (heights) drawn from each vertex to the opposite side in a triangle are concurrent, meaning that all three of these heights will meet at a single point inside (or outside) the triangle. This point is called the "orthocenter" of the triangle. 3. **Why It's Important**: The concept of公垂线 is important in geometric constructions and proofs. Understanding how the heights of a triangle work, and where they intersect, helps in solving problems related to the triangle's properties. ### Summary - **公垂线** refers to the lines drawn from each vertex of a triangle perpendicularly to the opposite side. - These lines are known for meeting at a common point (the orthocenter). - Understanding these concepts can help solve various geometric problems involving triangles. I hope this explanation helps clarify the concept for you! If you have more questions or need further clarification, feel free to ask!