Background
This is Problem 5-17 of John Lee's Introduction to Topological Manifolds.
Suppose $\sigma=[v_0,\ldots,v_k]$ is a simplex in $\mathbb{R} ^n$ and $w\in \mathbb{R} ^n$. If $\{w,v_0,\ldots,v_k\}$ is an affinely independent set, we say that $w$ is affinely independent of $\sigma$. In this case, the simplex $[w,v_0,\ldots,v_k]$ is denoted by $w*\sigma$ and is called the cone on $\sigma$. More generally, suppose $K$ is a finite Euclidean (geometric) simplicial complex and $w$ is a point of $\mathbb{R}^n$ such that each ray starting at $w$ intersects $|K|$ in at most one point. Define the cone on $K$ to be the following collection of simplices in $\mathbb{R}^n$: $$ w*K=K\cup\{[w]\}\cup\{w*\sigma:\sigma\in K\}. $$
Problem
Prove that $w*K$ is again a Euclidean simplicial complex, whose polyhedron is homeomorphic to the topological cone on $|K|$ (the topological cone $C|K|$ here is defined as the quotient space $(|K|\times [0,1])/(|K|\times\{0\})$).
My thoughts
First I have to check that $w*\sigma:\sigma\in K$ really is a simplex, which is to say that the union of $\{w\}$ and the set of vertices $\{v_0,\ldots,v_k\}$ of a simplex of $K$ is an affinely independent set. Suppose, on the contrary, that $\{w,v_0,\ldots,v_k\}$ is not affinely independent, where $\sigma =[v_0,\ldots,v_k] \in K$, then $w-v_0=\sum_{i=1}^{k}a_i(v_i-v_0)$ for some $a_1,\ldots,a_k$ not all zero. The ray is $x-w=b(x_0-w)\Rightarrow x=w+b(x_0-w)$, where $x_0\ne w,b\ge0$. We prove that the ray $x-w=b\left(\sum_{i=0}^{k}c_iv_i-w\right)$, where $\sum_{i=0}^{k}c_iv_i=x_0$ lies in the open simplex (all $c_i$'s are strictly positive) spanned by $\{v_0,\ldots,v_k\}$, intersects $\sigma$ for every $b=1+\varepsilon$ for any small enough $|\varepsilon|$. Take $$\begin{aligned} x &= w+(1+\varepsilon)(\sum_{i=0}^{k}c_iv_i-w)=(1+\varepsilon)\sum_{i=0}^{k}c_iv_i-\varepsilon w\\&=(1+\varepsilon)\sum_{i=0}^{k}c_iv_i-\varepsilon \left[v_0+\sum_{i=1}^{k}a_i(v_i-v_0)\right]\\&=\left[\varepsilon(\sum_{i=1}^{k}a_i-1)+(1+\varepsilon)c_0\right]v_0+\sum_{i=1}^{k}\left[(1+\varepsilon)c_i-\varepsilon a_i\right]v_i, \end{aligned}$$ where the coefficients of $v_i$ add up to 1, thus the point $x$ in the ray lies in $\sigma$ as long as $|\varepsilon|$ is small enough, a contradiction.
Next I have to prove that $w*K$ is a Euclidean simplicial complex. This is where I got stuck. I think only the intersection condition is nontrivial: take any pair of simplices, then their intersection is either empty or a face of each. I focused on the case where both simplices of the cone are of the form $w*\sigma _1,w*\sigma _2$, where $\sigma _1, \sigma_2 \in K$. My geometric intuition tells me that $(w*\sigma_1)\cap (w*\sigma_2)=w*(\sigma _1\cap \sigma _2)$, where $w*(\sigma _1\cap \sigma _2)$ certainly satisfies the condition. It is also trivial that $(w*\sigma_1)\cap (w*\sigma_2)\supseteq w*(\sigma _1\cap \sigma _2)$, but I failed to prove the opposite inclusion. Can you help me? I am aware of some related posts like (Is this a counterexample to problem 5-17 in John Lee's Intro to Topological Manifolds?) and ($\Delta$-complex structure of the cone and the suspension), but they don't seem to address my core issue.
