判断一个点是否在圆内（三点确定一个圆）

三角形的外接圆圆心是任意两边的垂直平分线的交点。 三角形外接圆圆心叫外心。

        /// <summary>
		/// Returns true if the point (p) lies inside the circumcircle made up by points (p1,p2,p3)
		/// </summary>
		/// <remarks>
		/// NOTE: A point on the edge is inside the circumcircle
		/// </remarks>
		/// <param name="p">Point to check</param>
		/// <param name="p1">First point on circle</param>
		/// <param name="p2">Second point on circle</param>
		/// <param name="p3">Third point on circle</param>
		/// <returns>true if p is inside circle</returns>
		private static bool InCircle(Geometry.Point p, Geometry.Point p1, Geometry.Point p2, Geometry.Point p3)
		{
			//Return TRUE if the point (xp,yp) lies inside the circumcircle
			//made up by points (x1,y1) (x2,y2) (x3,y3)
			//NOTE: A point on the edge is inside the circumcircle
            // 确定三个点是否组成三角形，三点共线则不构成圆
			if (System.Math.Abs(p1.Y - p2.Y) < double.Epsilon && System.Math.Abs(p2.Y - p3.Y) < double.Epsilon)
			{
				//INCIRCUM - F - Points are coincident !!
				return false;
			}

			double m1, m2;
			double mx1, mx2;
			double my1, my2;
			double xc, yc;
						
			if (System.Math.Abs(p2.Y - p1.Y) < double.Epsilon)
			{
                //计算边P2P3的垂直平分线的斜率
				m2 = -(p3.X - p2.X) / (p3.Y - p2.Y);
                //计算边P2P3的中点
				mx2 = (p2.X + p3.X) * 0.5;
				my2 = (p2.Y + p3.Y) * 0.5;
				//Calculate CircumCircle center (xc,yc)
				xc = (p2.X + p1.X) * 0.5;
                // 边P2P3与P1P2边的垂直平分线联立计算交点
                // y=xc 与 y=m2(x-mx2)+my2 联立得
				yc = m2 * (xc - mx2) + my2;
			}
			else if (System.Math.Abs(p3.Y - p2.Y) < double.Epsilon)
			{
                //计算边P1P2的垂直平分线的斜率
				m1 = -(p2.X - p1.X) / (p2.Y - p1.Y);
                //计算边P1P2的中点
				mx1 = (p1.X + p2.X) * 0.5;
				my1 = (p1.Y + p2.Y) * 0.5;
				//Calculate CircumCircle center (xc,yc)
                // 边P2P3与P1P2边的垂直平分线联立计算交点
                // y=xc 与 y=m1(x-mx1)+my1 联立得
				xc = (p3.X + p2.X) * 0.5;
				yc = m1 * (xc - mx1) + my1;
			}
			else
			{   
                //计算边P1P2和P2P3的垂直平分线的斜率
				m1 = -(p2.X - p1.X) / (p2.Y - p1.Y);
				m2 = -(p3.X - p2.X) / (p3.Y - p2.Y);
                //计算边P1P2和P2P3的中点
				mx1 = (p1.X + p2.X) * 0.5;
				mx2 = (p2.X + p3.X) * 0.5;
				my1 = (p1.Y + p2.Y) * 0.5;
				my2 = (p2.Y + p3.Y) * 0.5;
				//Calculate CircumCircle center (xc,yc)
                // y=m1*(x-mx1)+my1 与 y=m2*(x-mx2)+my2 联立
				xc = (m1 * mx1 - m2 * mx2 + my2 - my1) / (m1 - m2);
				yc = m1 * (xc - mx1) + my1;
			}
            //计算圆的半径
			double dx = p2.X - xc;
			double dy = p2.Y - yc;
			double rsqr = dx * dx + dy * dy;

			//double r = Math.Sqrt(rsqr); //Circumcircle radius
            //计算点到圆心的距离
			dx = p.X - xc;
			dy = p.Y - yc;
			double drsqr = dx * dx + dy * dy;
            //判断点到圆心的距离是否小于圆的半径
			return (drsqr <= rsqr);
		}

判断一个点是否位于圆内（圆由三个圆上的点确定）。

思路：

1. 圆上任意三个点不共线。

2. 圆上三个点构成的三角形的垂直平分线交于圆心，即三角形的外接圆。 

3. 直线方程的点斜式，y=k(x-x1)+y1。

4. 斜率的计算要考虑垂直于y轴的情况，即y1=y2的情况。