How Far Can You See from a Mountain? Tangent Lines on a Sphere

① This GeoGebra example demonstrates, in a 3D view, how far one can see from a mountain peak on a spherical Earth, by constructing tangent rays from the peak to the sphere and finding the horizon circle. ② Core mathematical concepts include spheres, tangent lines, points of tangency, the circle of the horizon, and the Pythagorean theorem applied to right triangles to compute the sight distance. ③ Construction highlights: use Sphere to create the Earth, Ray and Point for the sight lines from the peak, Tangent to find tangency points, IntersectPath to obtain the horizon circle, Circle and Segment for auxiliary elements, Angle to mark key angles, and Text objects to display numerical results. ④ Suitable for upper secondary or early university students learning solid geometry, tangent properties, and the Pythagorean theorem in real-world contexts, as well as for geography or astronomy outreach to illustrate the effect of Earth's curvature on visibility.