The accepted answer works great, but the resize logic ignores the case in which the image is larger than the maximum in only one of the axes (for example, height > maxHeight but width <= maxWidth).
I think the following code takes care of all cases in a more straight-forward and functional way (ignore the typescript type annotations if using plain javascript):
private scaleDownSize(width: number, height: number, maxWidth: number, maxHeight: number): {width: number, height: number} {
if (width <= maxWidth && height <= maxHeight)
return { width, height };
else if (width / maxWidth > height / maxHeight)
return { width: maxWidth, height: height * maxWidth / width};
else
return { width: width * maxHeight / height, height: maxHeight };
}