Geometric Intuition for Caratheodory's Theorem (for Convex Sets)

You basically add $$y = \sum_{j = 1}^k \lambda_j \, x_j$$ and $$0 = \sum_{j = 1}^k \alpha \, \mu_j \, x_j, $$ for some $\alpha \in \mathbb{R}$. This yields $$y = \sum_{j = 1}^k \underbrace{(\lambda_j + \alpha \, \mu_j)}_{=:\Lambda_j} \, x_j. $$ This directly yields $$\sum_{j=1}^k \Lambda_j = 1.$$ However, you additionally need $$\Lambda_j \ge 0 \;\forall j \qquad\text{and}\qquad \Lambda_i = 0 \text{ for some } i,$$ such that you obtain a convex combination, in which one coefficient is zero.

Now, try to figure out how to choose $\alpha$ and $i$.